#8976301

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4745 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

(2,5)(6,8)(9,10,15,13)(11,12,16,14)(17,18)(19,23)(20,25)(21,29)(22,24)(26,31)(27,30)(28,32);(3,7)(6,8)(10,13)(11,16)(17,19,22,27)(18,23,24,30)(20,26,28,29)(21,25,31,32);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,13)(15,16)(18,24)(19,27)(20,28)(21,31);(9,12,15,14)(10,11,13,16)(17,20)(18,25)(19,26)(21,30)(22,28)(23,31)(24,32)(27,29);(1,3)(2,6)(4,7)(5,8)(9,13,15,10)(11,12,16,14)(17,19)(18,23)(20,29)(21,25)(22,27)(24,30)(26,28)(31,32);(1,2)(3,6)(4,5)(7,8)(9,14,15,12)(10,16,13,11)(17,21)(18,26)(19,25)(20,30)(22,31)(23,28)(24,29)(27,32)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8883497);